Electrical computer



Jan. 19, 1954 c. J. HIRSCH 2,666,576

ELECTRICAL COMPUTER Filed May 11, 1951 5 Sheets-Sheet l SAMPLING GENERATOR CIRCUIT INVENTOR. FIG.3 CHARLES J. HIRSCH ATTORNEY Jan. 19, 1954 Filed May 11, 1951 c. J. HIRSCH 2,666,576

ELECTRICAL COMPUTER 3 Sheets-Sheet 2 n U al 6 22* I A 2 1 i 01- 3Q 2 I L. I W 22 g g 6 8 g i z i i m I O I l 2 1 1 I l.- I

o i l g i i I I 018' *2- 3 n 8* Txmea lg FIG.7 0

SAMPLING CIRCUIT 35 TIMING PULSE- PULSE GENERATOR GENERATORO CIRCUIT 2o COMPARISOBL/ CIRCUIT 4 INVENTOR.

CHARLES. J. HIRSCH ATTO R N EY Jan. 19, 1954 c. J. HIRSCH 2,666,576

ELECTRICAL COMPUTER Filed May 11, 1951 5 Sheets-Sheet 5 SAMPLING CIRCUIT PULSE- GENERATOR CIRCUIT COMPARISON CIRCUIT CIRCUIT I I O b PULSE GENERATOR 35 CIRCUIT J SAMPLING 27 95 96 TIMINQ sAw-TooTH o 97 I COMPARISON PULSE WAVE-SIGNAL 98 99 20/ CIRCUIT o GENERATO R GENER T T o 1 ls l llh V FIC -3.6

P V I INVENTCR v I CHARLES J. HIRSCH t1 t2 t3 4 ATTORNEY iatentecl Jan. 19, 1 95 4 Gharle's i. Hirsch; nouglastd'n; N; Y.',' assignbr to Hazeltine; Research, Inc.'; Chicago; 111;; a; cor

partition of Illinois Actuarial: may 1951; serial Na; 255,721

i5 (sizes-c1) 1 GENERAL ber 5, 1949; both entitled Electrical Computer! Onefl'general type of prior computer, which may be referred to? as a digital'computer, in cludes relay machines, punch-card machines, and adding and'multiplying machines utilizing either mechanical or electrical counting devices. Conventional computers of this type can handle numerical data after the; problem has; been reduced. to a; numerical routine susceptible to solu tions by' digitalmethods oftenrequiringextensive programming of the operation of the machine. These computers are inherently capable of very high accuracies. theaccuracy usually being limited only by the number ofgplaces to which a computation is carried out,-but the machinemay have to performa veryextensi ve counting opera;- tionto solve even a simple algebraic; expression. Conventional computers of this type tend; to be bulky andcumbersome in operation; particularly when the problem is at all complex as in the case ofmathematical series. a 1 I I Another type of prior \art computer maybe classifiedgenerally as a continuolllyqvariable computer. Computers of this type deal with quantities by continuousmechanical displacements or electrical eifects, Tachometer instruments come under this classification. These computers are known in the trade asanaloguecomputers: The electronic analogue computers use techniques which are similar to those conventionally used in 5 the field of communications whereas the digital computer' uses techniques similar to those .used in the fieldcf pulse-modulation;systems; Insofar as is known, computers of the"analog-ue" type a e not capable of solving equations involving mathematical series" without setting up a ccom purer for eacmerm of the series andthe'n' 'manu a lly' or bymeans of another computer adding 'the values" obtained: for each terms Such an ar= rangement is undesirable:

Compared with digital computers, the'an'a ldgu'e' computers usually have" the advantage of hi h speed and facility ofse'ttihg up the com p'uter" to? solve a given problem; but have the correlation with disadvantage that" their accuracy tends to ..b.e lower; Usually analogue computers arembre suitable for small scale problems that do. not justify extensive programming to provide .an approximate answer to such problems. Simple analogue computers capable of solving [mathematieal series" are unknown. If amore'elaborate arrangement as described above isxused, theniin spite of the size and complexity of the equipment utilized; such equipment is very limited in the parameter values which maybe used... v i.

It is" an object of the present invention, therefore; to providev a" new and improved: electrical computer which avoids one or moreoi. the limitations and disadvantages of prior comput'ers: r

Itis another object of theiinventiongtolprovide a new and improved electrical computer capable of solving equations involving: mathematical series I l '.r v. i

It is a further object of theinv'entio'n to provicie a n a d. im ed. ol o r l...comaut which d s t re ui e mohan o m in pa ts. is. ompac a d. i ht; .i wei t Ye si eb hls m nscomn ex c putat n at. h 515 i It s. a un. robio the event ov de anew. a i impr ved. oleotrig some capable. v C ntinu usl and. api ly alc at ,a I problem involving parameters subject to ar st l n er. obje t o th .iii et to ov d .newan p m rore el r a i o n for, p fo mi ma h ca o i 115 ii which. all at the nde d e an on i variables are 7 represented by voltages referredto a oo reniee r iommewue e- A,

a o "dan9e;w th h n en n r 'e i 1 computer for solving equations mvmvin'gia matical series including known and. unknown rame e s som r s meah Ts iil s l n electrical signal and a plurality of networks at e st m i...W 2 r e eoifi'o i' r s e ies. e on. vw ea h, ether aiiolswi e up yi st.. r ea1 v E h of tho n works has ir uit elements 150. p op ion d that th impede woes of .saidne work re. prop rti neiwnh respect; toeach'otherzin the ratio of. individual ones of one'group of components-of the 'termszof sai-d mathematical series" and the time constants of said networks are proportioned with'resp'ectto each other in the ratio of individual ones of another group ofcomponents of the 'terms ofsai'd mathematical-seriesso that upon applicatioriof the above=mentioned electrical signal to these networks-there are" developed-"therein electrical- In addition, the computer includes a signal-.

sampling circuit responsive to the control effect and coupled to at least one of the networks for determining the value of the aforesaid electrical effects developed therein at the aforementioned instant of time to derive a resultant electrical effect representative of the value of an unknown parameter of the mathematical series.

For a better understanding of the present invention, together with other and further objects thereof, reference is had to the following description taken in connection with the accompanying drawings, and its scope will be pointed out in the appended claims.

Referring now to the drawings, Fig. 1 is a circuit diagram of an electrical computer embodying the present invention in a particular form; Fig. 2 is a graph useful in explaining the operation of the Fig. 1 computer; Figs. 3 and 4 are circuit diagrams, partly schematic, of modified forms of the computer represented in Fig. 1; Fig. 5 is a circuit diagram, partly schematic, of another embodiment of the invention; Fig. 5a is a schematic diagram of a modified form of the computer of Fig. 5; Fig. 6 is a graph useful in explaining the operation of the Fig. 5 computer; and Fig. '7 is a graph useful in explaining the operation of the Fig. 50: computer.

Description of computer of Fig. 1

Referring now to Fig. 1 of the drawings, there is represented a circuit diagram of an electrical computer for solving equations involving mathematical series, particularly power series, including both known and unknown parameters. The computer comprises means for supplying an electrical signal, including an electrical motor ill energized in a conventional manner through a pair of terminals II, II and having a shaft l2 mechanically coupled to a cam l3. The supplying means also includes a switch I4 connected in series with a battery l5 and arranged to be controlled by the cam it. One terminal of each of the milliameters liia, IE1) and E50 is connected to the battery i5 through the switch It and through corresponding ones of variable resistors Ha, lib and He.

The computer also comprises a plurality of networks 5la5ile, inclusive, each in the form of parallel-connected resistor-condenser circuits at leastsome of which are electrically coupled in series relation with the signal-supplying means. The networks 50a and 56b are coupled in series between one terminal of the milliameter liia and the negative terminal of the battery l5. Related networks 580 and 50d are coupled between the battery terminal just mentioned and one of the terminals of the milliameter [Eb and the network 506 is coupled between one of the terminals of the milliameter I60 and the negative terminal of the battery [5. Each of the networks Eda-Jule, inclusive, has circuit elements, specifically a resistor and a condenser connected in parallel, so proportioned as, upon application of the aforementioned electrical signal thereto, to develop in each circuit an electrical effect varying as a predetermined time function which, at a predetermined instant of time, reaches a value representative of one of the terms of the mathematical series. The time function is preferably an exponentially decaying one, and the predetermined instant of time is one determined by an independent-variable known parameter of the mathematical series.

The circuit elements of the network 50a are a resistor 51a and a condenser 52a, the other networks including analogous arrangements of resistors and condensers, each designated by the same number as the corresponding element in the network 50a with letter suffixes corresponding to the letter whims of the network of which the element is a part. For reasons which will be explained more fully hereinafter, the impedances of these networks are proportioned with respect to each other in the ratio of individual ones of one group of components of the terms of the mathematical series and the time constants of the networks are proportioned with respect to each other in the ratio of individual ones of another group of components of the terms of the series. More specifically, these networks are proportioned to have time constants related to each other in the ratio of the exponents of the terms of the mathematical series to be solved and the condensers and resistors thereof have values which will develop different initial voltages related to the factorial terms in the series. The resistors Ho, Ho and He are each adjustable to permit preselected amounts of current to flow through their respective circuits.

The computer also comprises a source of potential having a value representative of a known parameter, the independent-variable known parameter, specifically at battery I8 adjustable to supply potentials of selectable values. The battery I8 is connected tc terminals I9, [9 of a comparison circuit 20 to be described more fully in the following paragraph.

The computer also comprises means jointly responsive to the potential of the source 18 and at least one of the aforementioned electrical effects at the aforesaid predetermined instant of time for developing a control effect, this means being, specifically the comparison circuit 20. The unit 20 includes a tube 2| having a cathode connected through a cathode load resistor 22 to one terminal of the battery i8 and a control electrode coupled to the other terminal thereof, both connections being made through the terminals I9, IS. The anode of the tube 2| is connected to a source of potential +3. The cathode of the tube 2! is coupled through a resistor 23 to the anode of a diode 24, this anode also being coupled through a condenser 25 to a terminal 26. The cathode of the diode 24 is coupledthrough one of a pair of terminals 55, 45 to one terminal of the network We, the other side of the network being coupled through another one of the terminals 45, 45 to one of the terminals I8, 59.

The computer also comprises a signal-sampling circuit 2'! of the bridge rectifier type preferably comprising four diodes 28, 29, 3B and 3! arranged in a conventional bridge rectifier circuit. One pair of diagonally disposed terminals of this bridge circuit is coupled to at least one of the networks and preferably connected across the networks Sta-50d, inclusive, through a pair of terminals 32, 32 and a load resistor 33. The resistor 33 has a storage condenser 43 coupled in paraiii therewith. The other pair or aiagdriaiiy disposed terminals of the bridge is connected through a biasing battery 54, a pairof terminals 34,, 34' and a pulse-generator circuit 35 to the outputterminal 26 of the comparison circuit 23.

' The pulse-generator circuit 35 is" of the block ing oscillator type and includes a vacuum tube 36 having an anode coupled through one of a pair of terminals 31 to the terminal 26 of the unit 20 and through one winding of a transformer 38 to a source of potential +B. Another winding of'the transformer 38 has one terminal thereof connected to the control-electrode of the tube 36 and the other terminal thereof coupled through a resistor 39 to a source of biasing potential -C. The terminal of theresistor 39 which is not connected to the source O" is connected to a pulseforming delay line 40, the remote end of which is open circuit'd. The cathode of' the tube 36 and one terminal of the delay line 40 are connectedto the other one of the terminals 31, 31, which terminal is also connected to the junction of the networks andjtc. The output winding of the transformer 38 is coupledithrough the terminals 4| 4! to the terminals 34, 34 in the signal-sampling circuit 21; If the signal applied to terminals 31, 37 is of insufiicient magnitude, 9; pulse amplifier may be included between these terminals and the anode circuit of the tube. 3S.

Operation of the computer of Fig. 1

I The operation of the computer of Fig. 1 will now be described with reference to the group of curves in the graph of Fig. 2 which represent the voltage decay of the networks Silo-50c, inclusive, with respect to time. In order that the curves of Fig. 2 may be shown aligned with reference to their time coordinates, the scale factors of the ordinates thereof are different. To this end, curve A has an ordinate scale factor of unity, curve C has an ordinate scale factor of one-quarter unity and curves B and D have ordinate scale factors of approximately one-tenth unity. Considering now the operation of the computer of Fig. 1, some time prior to the time n represented in Fig. 2, each cycle the motor In causes the cam i3 momentarily to close the switch [4 thereby car-1s ing current to flow in the networks 50d-50e, inclusive. Because of the values of the resistors and condensers each of the networks, each thereof is charged to a peak potential of a different value at the time h. For the solution of a specific problem to be discussed more fully hereinafter, the network 50a is charged to a potential of approximately 6.28 volts as represented by curve A of Fig; 2-, the network 5% is charged to a potential of approximately 81 volts as represented by curve B, the network 500 is charged to a potential of approximately 41 volts as repre- -sen-ted by curve C and the network 511d is charged to a potential of approximately 75.5 volts as represented by curve D. The network 50a may also be charged to a potential similar to that of the network SUa and may also be represented by curve parison circuit 20. A portion of this potential by the wen known cathode-follower action is effec-- tively developed across the cathode load resistor '22. Therefore, the difference in potential betwee-n 6 the voltage across the network 50c and the voltage across the resistor 22 appears across the series combination of the resistor 23 and the diode 24.

At the time h when the cam it permits the switch 14 to disconnect the battery l5 from the networks, the condenser in each network commences to discharge exponentiallythrough the resistor therein in the mannerrepresented by the respective curves of Fig. 2. When the voltage across the network We drops just below the potential across the resistor 22 in the unit 20 at time t2, the diode 24 conducts to develop across the resistor 23 a pulse of voltage having nega tive polarity. This latter voltage is applied through the condenser 25 to the pulse generator circuit at this time. The leading edge of this pulse initiates the generation of a single potential pulse of short duration in the pulse generator 35. The duration of the pulse generated in the unit 35 is determined by the time required for a pulse to travel a round trip in the delay line and be applied to the control electrode of the tube 38. This duration is selected to be just great enough to provide for adequate sampling of the voltage then present across the networks 500,-- ifld, inclusive, at and immediately afterthe time t2;-

The pulse developed in the output circuit of the generator 35 is applied through the terminals 3 3 3 with such polarity and magnitude as to overcome the biasing potential developed by the battery Accordingly, the bridge circuit including the tubes 283I, inclusive, is rendered conductive for the duration of the applied pulse with the result that substantially the sum of the voltages then existing across the networks Sta-58d, inclusive, is developed across the resistor 33 and stored in condenser 43. It will be seen that the voltage developed across the 'resistor 33 represents the difference in voltage across the networks a, 50b and the networks 50c, 50d, being representative of the algebraic sum of the potentials across the networks 50 a 50d, inclusive.

Therefore, if the voltage in each network is representative of a term of a mathematical series,

g3 05 97 an. E0 S111 Since in Equation 1 0 is required to be expressed in terms of radians and, in applicants computer,

is to be expressed in terms of volts the following equation is useful:

where E1=a voltage representative of one radian e=a voltage proportional to 0 in the same units as EneEi, and

K=a multiplication factor needed to compensate for the fact that the term e/E1 is unable to represent an angle greater than unity, that is, one radian.

By utilizing the relationships of Equation 2, Equation 1 may be expressed as follows:

1 l i '7'1( E1) a E,

In order to arrange the terms in the Equation 3 in a manner more suitable for use in the computer of Fig. 1, the following relationships may be employed: I

where i=the steady-state current in amperes flowing through the resistors of the resistor-condenser (RC) circuits of Fig. 1 just prior to the opening of switch l4, that is, just prior to the time i1. =the resistance in ohms in the RC circuits C=the capacitance in farads in the RC circuits,

and

t2=the elapsed decay time of an RC circuit from time zero, that is, time h. In Equation 4 Using the relationships in Equations as, inclusive, the Equation 3 may then be rewritten as follows:

L a-RL R101] l- When Equations 3 and '7 are compared and the relationships of Equations 4 and 5 are utilized, it is seen that:

K" K- tR =E F=ZR1 and. thus 11-1 R..= RI (9) Equation 7 in View of Equation 9 may now be expressed in the form:

Equation 10 may be utilized in the computer of Fig. 1 to solve mathematical series of the type represented by Equation 1.

Assume now that it is desired to determine the value of sin 0, where 0 is equal to 11', E0 is equal to 1, and K is equal to Zn. Then, in accordance with Equation 2 Thus, by utilizing Equations 10, 13 and 14:

I 2 3 S... Man 4;

6.28 l 6.28 (l e! 2 7! 2 Referring now to Fig. 1, as previously stated the networks EQw-J'Ee, inclusive, are individually proportioned to represent terms of the Equation 10 above. Networks Sta and Elle are proportioned to represent the first term of the Equation 10, specifically where i-R1=6.28 volts in accordance with Equation 14 and at the time the terms of the series in Equation 15 are added to determine the value of sin 7r. Since the networks Eila and 553s are, arbitrarily, the first networks to have their parameters determined, they may have any suitable time constant, and any suitable value of the resistors fiia and tie may be utilized. For simplicity, a time constant of 10- seconds and a value of 1,000 ohms for the resistors tla and tile are selected, the condensers 52a and 52e thereby being determined to have values of one microfarad;

Since the Voltage developed across the resistor 5m should initially equal 6.28 volts, the resistor lid is adjusted to, permit a current of 6.28 milliamperes to flow through the meter Mia and, therefore, through each of the resistors 51a and 5422. Similarly the resistor [Tb is adjusted to cause a corresponding amount of current to flow through the resistors die and 5563 though it is to be understood that if suitable scale factor corrections are made, the current flowing in the resistors 55c and Bid may difier from that flowing in the resistors 5m and 511). For scale factor purposes and in order that the operation of the comparison circuit 26 may be less critical and therefore more accurate, it may be desirable to make the voltage scale of the battery it and across the network 592 one-twentieth of the voltage scal utilized in the network 58a. Thus, whereas the network :Eiia may have a voltage equal to 6.28 volts therevoltages in the networks Elia-50d, inclusive, are to be .added when the network 58a has decayed to have a voltage equal to one-half 6.28, the battery i8 .is adjusted to have a voltage of one-half 125.6 volts, or 62.8 volts, this being one-half of the initial voltage across the network 50c.

Having selected the time constant and resistor values for the network 59a representing the first term of the series defined by Equation 15, the corresponding values of the networks Silly-55d, inclusive, are determined by Equation 10 above. Specifically, the time constants of the networks Sou-50d, inclusive, are arranged to be in the ratio of the powers of the exponential component of the terms which the networks represent. The network 500 representing the second term of the series is proportioned to have a time constant one-third that of the network 50a, since th ex- .ponential term of the second term of the series is indicated as being raised to the third power. In a similar manner the network 5017 representi-ng the third term and the network 50d representing the fourth term of the Equation 15 are proportioned to have, respectively, time constants one-fifth and one-seventh that of the time constant of the network 50a. Equation 9 determines the values of the resistors 5-lb'5ld, inclusive, based on the value of the resistor 5| it. Thus the resistor 5| is:

g =6,550 Ohms (16) Similarly, the resistor 51'?) equals 12,900 ohms and the resistor Id equals 12,200 ohms. The values of the condensers 52b52d, inclusive, are determined in the conventional manner by the selected time constants and resistor values for the networks.

Having proportioned the networks Sta-Que, inclusive, in the manner just described, these networks now represent four terms of the power series for the sine functions, specifically of the series for sin 1r. In the manner previously described, the networks 5-Oct- 502, inclusive, are charged by potentials obtained from the battery l5. As represented by Fig. 2 the initial voltage across the network 56a is approximately 6.28 volts; across the network 59b is approximately 81 volts; across the network 500 is approximately 41 volts; and across the network 513d is approximately 7.6.5 volts, these voltages being determined 'by the relative values of the resistors in the net- Works, Because of the scale factor diiierence, the initial voltage across the network Elie is, as has been previously stated, 125.6 volts. t2 when the voltage across the network 562 is equal to the voltage across the battery it, specifically 62.8 volts, the voltage across the network 50a is equal to 3.14 volts and is therefore representative of the first term of Equation 15. Also at the time is, the voltage across the network 55?) is equal to the value of the third term, the voltage across the network 500 is equal to the value of the second term and the voltage across the network 50d is equal to the value of the fourth term of the Equation 15. As indicated by the Equation '15 the voltages in the network 5011 representing the first term and the voltage in the network 591) representing the third term should be added,

while the voltages in the network 500 representing the second term and in the network 5M representing the fourth term should also be added, the latter sum being substracted from the first- :mentioned sum. This "operation is effected by arranging the networks in the manner previously At the time the first-mentioned inductor.

described so that the sampling circuit 21 develops across the resistor 33 the algebraic sum of the voltages in the networks a--5lld, inclusive. It will be seen that this sum is the same as that determined by solving the Equation 15 in a conventional manner for the sine of 11'.

Description of computer of Fig. 3

There has previously been described a simplified arrangement for solving mathematical problems involving series functions in which, for purposes of simplicity of explanation, the networks having characteristics individually related to the terms of the series have been described as having predetermined time constants and resistor and condenser values. It should be understood that for purposes of flexibility in solving problems of the type under consideration, such networks do not usually have predetermined fixed characteristics but rather include adjustable elements so that any desired time-constant value and any desired impedance value for a network may be obtained. In this way many diiferent problems may be solved. Fig. 3 represents a computer including a plurality of inductor resistor networks having greater flexibility in the electrical characteristics obtainable therefrom than the resistor condenser networks of the computer of Fig. 1. In describing and explaining the operation of the computer of Fig. 3, since the computers of Fig. 1 and Fig. 3 are related, similar components thereof are designated by the same reference numerals and analogous components by the same reference numerals with a factor of 300 added thereto.

There is included in the computer represented by Fig. 3 of the drawings, a timing pulse generator 655 which may be a multivibrator for periodically developing electrical signals and may be adjustable in the frequency of operation thereof. The output circuit of the generator is coupled to control electrodes in a pair of electron-discharge devices, specifically tetrode tubes GI and 52. The tubes 6| and 62 each have the anodes thereof connected to a source of potential +13 and the screen electrodes thereof individually and respectively coupled to batteries 3l'5a and Mob. The cathodes of the tubes 6! and 62 are coupled through load resistors Ila and He, respectively, to inductor resistor networks 3.590 35lld, inclusive, and 350e, respectively. Networks 35tla-35lld, inclusive, are connected in series between the load resistor Ila and ground while the network 35Ile is connected between the load resistor I10 and ground. Each of the networks 350a350e, inclusive, includes a resistor connected in parallel with a variable inductor and having another inductor inductively coupled to The amount of coupling between the inductors is adjustable to effect any desired amount thereof, thereby to proportion the magnitudes of the impedances of the networks with respect to each other, as viewed from the terminals on a terminal block 64, in the ratio of the coefficients of the terms of the mathematical series. Specifically, the network 350a includes a resistor 35m and an inductor 352a having an inductor 63a inductively coupled thereto and the networks 350b350e each include analogous arrangements of resistors and inductors each designated by the same number as the corresponding element in the network 350a with letter suffixes corresponding to the letter suffixes of the network of which the element is a part. Each of the secondary inductors 630-4311, inclusive, is connected toa 1 1 pair of terminals on a terminal block 64. For example, the inductor 63a is connected to terminals 55a, 65a and the other inductors are connected to other pairs of terminals designated by similar numerals and having letter sunixes corresponding to the associated network. There is also coupled to a pair of terminals 66, 66 on the .terminal block (it a circuit including a battery 81,

a variable resistor 68 and a load resistor 69 connected in series, the terminals 66, 65 being connected across the load resistor 69.

The terminal block 64 includes an output terminal i2 and may be of any conventional type permitting connections to be made from any terminal to any other terminal thereon. switchboard type arrangements may be employed and, if desired, an elaborate switching system may be used in place of the plugs and jacks on a switchboard. The output terminal !2 of the terminal block 54 is effectively coupled through a doublepole double-throw switch either to the input circuit of the sampling circuit 2'! or of the comparison circuit 20. Similarly the secondary winding 83c of the network 3551c is coupled through the switch It! either to the unit 21 or the unit 26. The switch Hi is so arranged that when the output terminal 12 of the terminal block 54 is coupled to either the unit 21 or the unit 29 the winding 63c is coupled to the other thereof.

Explanation of operation of the computer of Fi 3 Considering now the operation of the computer of Fig. 3, as previously explained with reference to the computer of Fig. 1, the networks ilameans of the tubes 6| and 62 and the batteries 3i5a and N52), to cause current to flow in each of the inductor resistor networks SSW-350e, inclusive. By means of the terminal block 64 the voltages across the difierent secondary windings 63a63d, inclusive, may be combined additivel e or subtractively by appropriate connections of the terminals of one secondary winding to those of another secondary winding. Thus to solve the Equation 10, previously solved by the computer of Fig. l, the secondary windings 63a and i 53b are so connected that the voltages developed therein are additive as are the secondary windings 63c and 6301 but the group including the windings 53a and 63b is connected so that the voltages developed therein combine subtractively with the voltages developed in the group including the windings 63c and 63d.

The circuit including the battery 51 and the resistors 63 and 69 provides a means for develop ing in the output circuit of the computer networks a fixed potential related to a fixed factor which might be present in the series function to be added to the terms thereof. Thus in the solu tion of the series for the cosine of an angle the first term thereof is a fixed term and might be represented by the voltage developed across the terminals 66. 66.

The double-pole double-throw switch Hi provides a means for arranging the computer to solve for the sum of a series as described with reference to the computer'of Fig. l or, knowing the sum thereof, to solve for the value of any term of the series or for the value of the variable in each term of the series. Thus the value of sin 0 may be determined or knowing the value thereof the angle 0 may be found. When the switch it is in one operating position so as to couple the output circuit of the terminal block @4 to the sampling circuit 2'5 and the secondary winding 63c to the comparison circuit 253, the solution of the series is as previously described with reference to Fig. 1. When the switch ii) is in the other operating condition thereof the output circuit of the terminal block 6 1 is coupled to the comparison circuit 2% and the winding 53c is coupled to the sampling circuit 2?. It is seen that in this position when the comparison circuit determines that the voltages across the networks Elia- 53d, inclusive, represent the sum of a certain series, for example, sin 11', the potential across the sec ondary winding tide is sampled and represents the term 11'. In this manner knowing the sum of the series but not knowing what angle the series represents, the value of the angle may be determined;

Description and explanation of operation of the computer of Fig. 4

The computer of Fig. 4 is related to the computer of Fig. 3 but utilizes resistor condenser networks instead of inductor resistor networks. In addition, the arrangement of the networks of Fig. i is similar to the arrangement described with reference to Fig. 1. Therefore, with reference to Fig. 4, components and elements similar to those of Figs. 1 and 3 are represented by the same reference numerals and analogous components by the same reference numerals with the factor of 4:00 added thereto with respect to the units of Fig. 1.

Each of the networks ta450d, inclusive, includes a pair of variable resistors connected in series and a condenser connected in parallel therewith. Thus the network 450a includes a condenser 52a and a pair of variable resistors a and em. The same referencing system previously used with respect to the elements of the networks of Figs. 1 and 3 is also used with re spect to the resistors and condensers in the networks 45iia45ad, inclusive, of Fig. 4. The resistors ale-Bid, inclusive, are adjustable to determine the time-constant characteristics of the diiferent networks. The resistors Stine-86d, inclusive, are adjustable to determine the relative initial voltages to be developed across the diiierent networks and, therefore, are adjusted in the different networks to represent the coefficient of the term represented by the network.

Considering now the operation of the computer of Fig. 4, it is seen that the time constants of the networks are such that the networks individually develop voltages varying as predetermined time functions individually related to the exponent of a term of the series. The resistors Eda-89d, inclusive, are adjustable to relate the networks diiiia45l3d, inclusive, in a manner representative of the relationship of the coefiicients of the terms of the series. The networks d5ila45iid. inclusive, are energized by the timing pulse generator St; in the manner described with reference to Fig. 3.

The network 45M in the computer of Fig. 4 serves the dual purpose of developing a voltage representing one of the terms in the mathematt cal series, which term is to be added to the other axe -me terms thereof to obta'm the sum of the series, and also representing the control voltage the value of which at some time activates the comparison circuit 2 to effect the adding of the terms. The possibility of an arrangement of this type was discussed with relation to the computer of Fig. 1.

The computers represented by Figs. 1., 3 and 4 have been described as being capable of solving power-series type equations and an explanation of the manner in which such computers solve equations of this type has been presented. The equation utilized as an example is one in which the exponents of the terms thereof are integers.

It should be "understood that computers of the type described in accordance with the present invention are not limited to solving equations having such exponents. Power-series equations 'having exponents which are not integers may be solved with equal facility. Thus an equation of the type .may be solved by properly proportionin'g the time constants of the networks relative to one another to represent the relative values of the exponents of the different terms of the equation, as shown in Equation 6a.. Therefore, the exponents may be either integers or nonintegers. By utilizing equations having noninteger exponential terms, fewer terms of the equation may be required to ascertain the convergence of the equation.

Description of computer '07 Fig.

The computers discussed previously herein are designed ,primarily to solve power-series type equations. The computer now to be described and represented by Fig. 5 is designed to solve Fourier-type equations. Since many of the components of the computer of Fig. 5 are related to components in the computers of "Figs. 1 and 3, similar ones thereof are designated by the same reference numerals and analogous ones by the same reference numerals with a factor of 500 added to the reference numbers used in Fig. 1 for such units.

In the computer of Fig. 5, the shaft H of motor Ill includes a plurality of cams l 3a-l3d, inc'lusive, mechanically coupled thereto. The cams l3ai3c, inclusive, are arranged to close a plurality of switches [401-440, inclusive, at the time the cam I30! opens a switch Md.

The computer of Fig. 5 also includes a plurality of resonant networks each having input and output circuits and arranged in a plurality of groups. At least the input circuits of the networks 'of at least one of these groups are electrically coupled in series relation. Thus the networks 550a550c, inclusive, each including inductively coupled inductors, having a condenser in parallel with one of the inductors, comprise one group, the input circuits of the networks of which are coupled in series through the resistor Ila, the switch 14d and a battery 545. At least the input circuits of the networks of another of the above-mentioned groups are electrically independent of each other. Thus networks 99'a-90c, inclusive, each also including inductively coupled inductors, comprise another one of these groups, each network thereof :including a closed'circuit of an inductor, a condenser, a battery and one of the switches Mir-I40, inclusive. 7

The motor In, the cams Isa-43d, inclusive,

the switches i4al4d, inclusive, together with ergizin'g .means.

the battery 5-!5 and the battery in each of the networks a- 9flc, inclusive, comprise an en- This energizing means is arranged collectively to energize the networks :ssea -s5-oc, inclusive, at one instant of time by the operation of the switch [4d and individually to energize the networks tim 90e, inclusive, at a later instant of time by the individual operations of the switches Ida-44c, inclusive.

Each-of the networks 55911-5500, inclusive, and 9060 990, inclusive, have circuit elements, specifically the condensers and inductors thereof, so proportioned and the coupling between the inductors so arranged that upon being energized there is developed in each network an oscillating signal, the oscillating signals in the networks 155i!d--550e, inclusive, being out of phase with the oscillating signals in the networks 9Ba90c, inclusive. Individual ones of the networks 55Ba-550c, inclusive, and Silo-90c, inclusive, :ar'e proportioned to have different resonance characteristics so that the signal developed in each network has a frequency peculiar to the developing network. More specifically, as will be better understood hereinafter, the resonance characteristics of the networks 55fla 55flc, inelusive, and eta-c, inclusive, are related to each other in the ratio of the angles of the sine and cosine terms in the mathematical series to be solved and the coefficients of coupling of the inductors in each network are related to each other substantially in the ratio of the coefiicients of the terms of the mathematical series. Stated more generally, since a resonant circuit .is a time-constant circuit having a complex time constant of e w that .is, the magnitude of a voltage therein varies with time as defined by e"""ci the time constants of such networks are proportioned with respect to each other in the ratio of individual ones of a group of components of the terms of the mathematical series, that is, in the ratio of the angles of the sine and cosine terms. The ir'npedi'ince's of such networks are proportioned by adjusting the coefficient of coupling of the inductors as described.

As in the computer of Fig. 3, the terminal board 64 is arranged to couple the output circuits of the networks 550a55llc, inclusive, and the networks 90a-90c, inclusive, in any desired manner. The connections represented on the board 54 in Fig. 5 relate to the coupling desired for the solution of a problem to be discussed hereinafter.

Explanation of operation of computer of Fig. 5

Considering how the operation .of the computer of Fig. 5, and referring "to the curves of Fig. 6. upon closin the switch Md at a time 151. the current flowing through the networks Stine-55cc, inclusive, slowly increases to a value determined by the voltage of the battery 515 and the resistor Fla. The voltage across each .of the networks Bianca-55cc, inclusive, and, therefore, across the secondary inductors '63a63c, inclusive, is essentially zero because of the slowness with which the current builds up to its final value, most of the voltage being developed across the resistor Via. Upon opening the switch Md at a time t2, the current in each of the networks sec-Paste. inclusive, cannot immediately drop to zero clue to the-inductance in these networks and, therefore, the condensers 52a -52c, inclusive, are charged. This charging results in producing in each of the networks Om-550e, inclusive, an oscillation at a frequency determined by the parameters of the individual networks to develop signals in the inductors B3a--63c, inclusive, having relative amplitudes determined by the impedances of the different networks and the coefficients of coupling of the inductors in each network. The signals developed in the networks Edda-55th:, inclusive, assuming that fundamental and second and third harmonic signals of equal amplitudes are developed in individual ones thereof, may be represented respectively by the curves AC, inclusive, of Fig. 6.

Also, at the time 162, the closing of the switches Ma,l4c, inclusive, shock excites the networks eta-90, inclusive, into oscillation at frequencies determined by their parameters to develop signals in the inductors Std-63), inclusive, having relative amplitudes determined as described with reference to the inductors t3a53c, inclusive. The signals developed in the networks sea-41cc, inclusive, assuming that these networks have the same relative relationships as those described with reference to the networks t tled-55cc, inclusive, may be represented respectively by the curves D-F, inclusive, of Fig. 6. It is seen that while the signals represented by the curves AC, inclusive, start from zero potential, the signals represented by the curves DF, inclusive, start at a maximum potential because the full battery voltage in each of the networks tiiw-Qdc,

inclusive, is applied instantaneously across the inductors in these networks, the condensers filo-Sic, inclusive, being unable to charge instantaneously. It is also seen that the curves A-C, inclusive, are related to the curves D--F, inclusive, in the same manner as sine and cosine functions are related, the curves D-F, inclusive, being out of phase with the curves A-C, inclusive, as a cosine function is out, of phase with a sine function.

The computer of Fig. 5 may be utilized to solve functions involving Fourier series-type relations. Thus the computer of Fig. 5 could be utilized to determine the harmonic composition of a saw-tooth wave defined as follows:

where Ezthe maximum amplitude of the saw-tooth wave in volts, and 1 x=the measurement along the :1: axis or ordinate of the wave in radians.

tive coupling between the inductors would be.

determined by the coeflicients of the terms in the equation. Also, the networks representing the first and third terms in the equation would be combined additively while the network representing the second term would be combined,

subtractively in the manner previously described with reference to Fig. 3. Thus, assuming that the networks 55% and Elite represent the first and third terms of Equation 18 and the network 5501) represents the second term thereof, the;

16 output circuits of these networks through means of the terminal board 64 would be combined as indicated in Fig. 5. The solution of an equation of the type represented by Equation 18 can, therefore, be readily understood.

If cosine terms were involved in the equation, one or more of the networks eta-99c, inclusive, would be required and the output circuits thereof would be so coupled with the output circuits of the networks 55960-5580, inclusive, in the terminal board t l that the sum thereof would represent the sum of the terms in the equation.

The manner in which the units 28, 35, 2'5 and the switch it are employed is fully described with reference to the computer of Fig. 3.

Description and explanation of operation of portion of computer of Fig. 5a

The computer of Fig. 5 as has been previously stated, is arranged to solve Fourier-type equations. It is limited in being able to solve only for the sum of the series when the value of one term thereof is known or for a term when the value of the sum-is known. As represented by Fig. 5 the computer is unable to solve for the sum of the series for any given value of an angle of one of the terms in the series. The arrangement of Fig. 5a does not have this limitation.

In 50', units similar to those of Fig. 5 are designated by the same reference numerals. In addition to such units, the arrangement of Fig. 5a includes a single-pole double-throw switch 93, one of the stationary terminals of which is connected to the terminal board 6 3, to one of the terminals 650 thereon, and the other of which is connected to an output circuit of a zero-adjusting circuit 94. The movable contact of the switch 93 is connected to a terminal of the double-pole double-throw switch It. A timing pulse generator to of a type previously described herein with reference to Fig. 3 and a saw-tooth wavesignal generator are connected in cascade with the input circuit of the unit 94. The saw-tooth wave-signal generator 93 may be of conventional type, a suitable circuit being disclosed in Principles of Radar published by McGraw-Hill Book Company, Inc., New York, New York (second edition 1946), page 3-20, Fig. 10. The unit 94 provides for the initiation of the saw-tooth wave at zero voltage at a given time and comprises a coupling condenser 97 in series with a resistor 38 across the input circuit of the unit 9 3. The resistor 98 has coupled in shunt thereto a diode 93 having a grounded anode.

Referring now to Fig. 7 there is represented by curve A, a sine wave similar to that represented by curve A in Fig. 6 and developed in the output circuit of one of the networks described with reference to Fig. 5. The saw-tooth wave-signal generator 86 has parameters so proportioned that the potential of the saw-tooth wave developed therein and represented by curve B of Fig. 7 has zero potential at the time t2, the time of the initiation thereof and of the initiation of the sine wave represented by curve A. At the time ta the saw-tooth wave has a potential representing the angle of in the sine wave represented by the curve A. Similarly, the saw-tooth wave is arranged to have potentials at other related times individually representing different angles of the sine wave. As a result, if a highly linear sawtooth wave is developed, the potentials therealong between Zero and the maximum potential at time in, will represent angle between 0 and 360. If, therefore, the switch 93 is connected scram ,tc the output circuit or the aero-adiustlns cult 9C" and the blades; or the switch Ht are closed in contact with the upper set of terminals thereof, there is continuously-"developedacross the input circuit of the comparison circuit 20, a potential representative of angles between 6 and 360. Therefore, in the -manner previously-described, thecomparison circuit m bysuitable adjustmerit.v of the battery is, maydetermine the sum of the series developed inthe networks of the computer for any given angle between and 360-: This sum, in: the manner previously. described, is represented'as a potential in the out put circuit of the sampling circuit- 21%.

The timing pulse generator 8! actsin aconventional mannerrepetitively to develop-a sawtooth wave of the type just described for every full cycle of the determining sine wave-as repre sented by curve A of Figffi". The zero-adjusting circuit 84 assures that the saw-toot-hwave'developed in the generator 96 will always'start-from zero potential,

There have been described hereincomputers in accordance with the present invention whichare capable of solving alltypes of mathematical series problems and it; is to be understood" that such computers are essentially mathematical: function generators. Such computers are not only capable of" solvingv series functions of ithe conventional type. but: can also solve an equations capable of bein expressed in the; form of a series function:- It, is." in this broader Sense that the term "ma in e matijcal series is utilized herein.

While there have been. d'escribedcwhat are at. present considered tohe. the preferredv embodiments of this invention, it will beohvious; to those skilled in the. are that. various change and modifications may be made; therein without departing, trom, the invention, vand-it is, therefore, aimed in the appended claimsto cover all such changes andmodificationsas fallwi-thin the tru spirit an scope: of the invention. a

What is claimedis: I

1. An electrical computerfor-solving equations having known and unknown parameters and involving mathematical: series including terms each. having; an exponent anda multiplier and comprising a signalegenerating, circuit for; periodically supplying an electrical signal; a, plurality of time-constan tnetworks al 196% some orwhich are electricallr coupledrin. se ies I ll tion, with-said si nal-g neratin c rcuit and; thereof having circuit; elements-so proportioned that the impedancesioif saidnetworksare pros portioned with respect to each other in the ratio of said multipliers of said terms and the time constants; of said networks are: proportioned with respect to each other in the ratia or said expo? nents. of said terms andimpedance characterise so that; .uporr application oi? said-electrical signal thereto, thereis: developed in; each thereet a different voltage varying as apredetermined time function which, at an instant or timedee termined by an independent-variable known parameter of said mathematical series-reaches a value representing" one: or the terms of said mathematical series; a source of potential havl ng auvalue; representative of saidindependente variable known parameterr-means for utilizing said potential or'said sc ircevandi atlleastf once! said voltages at: said-instant or said timej-to de-E velop. a; controlzeffect: and: a signer-sampling circuit coupled to saidutilizing means and to.

saidcontrol efiect for determining the value of; at least one of said voltages at=said;instant; of time to derive aresultant voltage representative of the value of an unknown parameter of said mathematical series;

' 2; An electrical computer for solving equations involving mathematical series including known and unknown parameters comprising;

, means for supplying an electrical signalga plu rality of networks at least some of which are electrically coupled in series; relation with each other d th ai supp n means a d each havijne c t el en s o proport oned hat he impedance of s d twor s ar ro o tioned with respect to each other inthe ratiQ Of: L111: vidual' ones of one group 01 components of} the terms of said mathematical series and the time constants of said networks are, proportionedwith respect to each other in the ratio of, individual ones. ofa h oup of c m o e t f the terms, of said mathematical series so that, upon app c n of. said e ct ical si nal to said iceworks there are developed therein electrical effects varying as predetermined time functions which ata predetermined instant of; time reach in individual ones of said networks: values repe resentative of different ones of the terngsqf said; mathematical series; a source of potential having a value representative of a l known parameter at said predetermined instant of time; means responsive jointly to said potential 01': said source and at least one of said electrical eiiects atsaid; predetermined instant of time for developinga control eif'ect; and a signal-sampling circuit responsive to said.- control eflfect, and coupled; to, at least one of said networks 1101f d'eterminingthe value of said electrical effects developed therein at sa Predetermine i ns ant of time v thereby to derive a resultant electrical effect represents; tive of the value of an unknown parameter of said mathematical series- 8. An electrical computer for solving equations involving mathematical series including known and unknown parameters and terms having coefieients and exponents comprising:

means for supplying an electrical signal; a; phira-l ity or resistor-condenser networks at least some: of which are electrically coupled in series relation withsaid supplying meansancl the magnitudesaof theresistors and condensers of; said networks being sohproportioned thatthe impedances oi said networks are proportioned with respect to, each other in'the-ratio of the coefficients of'the terms of said mathematical; series a m-cl the, time. constants of said networks are proportioned with respect toeach other in the ratio of the exponents of the terms of said series so that upon application of saidelectrical signal to said networks there are developed therein electrical effects varying as predetermined time functions-which at, a predeterminedinstantof timereach in individual onesef-said networks values:- representative of different onesof-the terms of said mathematical series; a source or potential having a value representative of a known parameter at said predetermined instant of time; means responsive jointly to said potential of said. source and at least one of saidw'elecfirical'fefiects. at. said predeterminedhinstantof time. fordeveloping; a control eifiect; and a signal sampling circuit coupled to said. responsive means atleast one of'said network e13 deterhe the value of at, least one of aid QIQG: trical effects" at said predetermined instant of atlmstflneof-sgid nefimmrmdlre ppn fye w 15 time, therebyto derive a resultant electrical effect representative of the value of an unknown parameter of said mathematical series. 4. An electrical computer for solving equations involving mathematical series including known and unknown parameters and terms having coefficients and exponents comprising: means for supplying an electrical signal; a plurality of inductor-resistor networks at least some of which are electrically coupled in series relation with said supplying means and the magnitudes of the resistors and inductors of said networks being so proportioned that the impedances of said networks are proportioned with respect to each other in the ratio of the coefficients of the terms of said mathematical series and the time constants of said networks are proportioned with respect to each other in the ratio of the exponents of the terms of said series so that upon application of said electrical signal to said networks there are developed therein electrical effects varying as predetermined time functions which at an instant of time determined by an independent-variable known parameter of said mathematical series reach in individual ones of said networks values representative of different ones of the terms of said mathematical series; a source of potential having a value representative of said independent-variable known parameter; means responsive jointly to said potential of said source and at least one of said electrical effects at said instant of said time for developing a control effect; and a signalsampling circuit coupled to said responsive means and to at least one of said networks and responsive to said control eifect for determining the value of at least one of said electrical effects at said instant of said time, thereby to derive a resultant electrical effect representative of the value of an unknown parameter of said mathematical series. I

5. An electrical computer for solving equations involving mathematical series including known and unknown parameters and terms having coefiicients comprising: means for supplying an electrical signal; a plurality of inductor-condenser networks at least some of which are electrically coupled in series relation with said sup plying means and the magnitudes of the inductors and condensers of said networks being so proportioned that the impedances of said networks are proportioned with respect to each other in the ratio of the coefficient of the terms of said mathematical series and the resonant frequencies of said networks are proportioned with respect to each other in'the ratio of individual ones of another group of components of the terms of said series so that upon application of said electrical signal to said networks there are developed therein electrical effects varying as predetermined time functions which at an instant of time determined by an independentvariable known parameter of said mathematical series reach in individual ones of said networks values representative of different ones of the terms of said mathematical series; a source of potential having a value representative of said independent-variable known parameter; means responsive jointly to said potential of said source and at least one of said electrical effects at said instant of said time for developing a control effeet; and a signal-sampling circuit coupled to said responsive means and to at least one of said networks and responsive to said control effect for determining the value of at least one of said electrical effects at said nst t 9f sa time, thereby to derive a resultant electrical effect representative of the value of an unknown parameter of said mathematical series.

6. An electrical computer for solving equations involving mathematical series including known and unknown parameters comprising: means for supplying an electrical signal; a plurality of networks at least some of which are electrically coupled in series relation with each other and with said supplying means and each having circuit elements so proportioned that the impedances of said networks are proportioned with respect to each other in the ratio of individual ones of one group of components of the terms of said mathematical series and the time constants of said networks are proportioned with respect to each other in the ratio of individual ones of another group of components of the terms of said mathematical series so that upon application of said electrical signal to said networks there are developed therein electrical effects varying as predetermined time functions which at an instant of time determined by an independent-variable known parameter of said mathematical series reach in individual ones of said networks values representative of different ones of the terms of said mathematical series; a source of potential having a value representative of said independent-variable known parameter; means jointly responsive to said potential of said source and at least one of said electrical effects at said instant of said time for developing a control effect; and a signal-sampling circuit responsive to said control effect and coupled across a plurality of said networks in series for determining the value of the sum of said electrical effects developed therein of said networks in series at said instant of said time, thereby to derive a resultant electrical effect representative of the value of the sum of the terms of said mathematical series.

7. An electrical computer for solving equations involving mathematical series including known and unknown parameters comprising: means for supplying an electrical signal; a plurality of net-- works at least some of which are electrically coupled in series relation with each other and with said supplying means and each having circuit elements so proportioned that the impedances of said networks are proportioned with respect to eachother in the ratio of individual ones of one group of components of the terms of said mathematical series and the time constants of said networks are proportioned with respect to each other in the ratio of individual ones of another group of components of the terms of said mathematical series so that upon application of said electrical signal to said networks there are developed therein electrical effects varying as predetermined time functions which at an instant of time determined by an independent-variable known parameter of said mathematical series reach in individual ones of said network values representative of different ones of the terms of said mathematical seriesia source of potential having a value representative of said independent-variable known parameter; means jointly re sponsive to said potential of said source and at least one of said electrical effects at said instant of said time for developing a control effect; a switching circuit having at least two operating conditions; and a signal-sampling circuit coupled through said switching circuit to said responsive means and to at least one of said networks and responsive to said control effect for 212 nstant pf said time. theretivoltaserepresentative .of

, flerge y t,,,r1vea eui tee sunlsnown parameter ofsaid esentat1v' of the va ue of an atioalseriesl eter ,of said mathemdiifia ser es. s 10;; on. electrical computer "for solving equa- 8. An electrical computer-for solving equatlons -;;tions*having-;known 1and nnknownpparameters i vo ving mat em ilta ser s nclu in kn w and imvoivme mathematical series includ n and unknown parameters comprising: gleam :fq ztumdcaoh havihg aniexponent iand armultiplier supplying i e ec rical-signal; a pluralityof net- -;.and':-'O0mprising:ha signal-senerating -.circ-uit for wor s rr nge apluralityof groups, those-in i0 periadicallvsnpplvingan electricalsignal; a plu each of i gr p being ele tr cally coupled in 'rality or time-constant networks at ,least some seriesrelation with .eachother and withsaid supof whicheare electrically coupled in series relajplyins mea and each netwo k havi .oireuit aionvwitheech other and with-said Signal-gent emen sso pr portioned that-the impedances of s ati n oircuit'i-and each thereof includin at said networks a e -.p portiened with resp ct t ,l; i lfidst iindliqmlid'iinduetively coupled in pairs and ea 9 1 1 n the-ra io of -'individual ones of: one circuit-elements including said inductorsso. :progroup of components of theterms of said v-mathepetitioned that-the =eceificients' of coupling of natical series and the time constants of .said said pair-sot;saidrindnctors of said networks are networks are proportioned with respect to "each proportioned -'with respect to each other in the other in the ratio of individual .onesof' another go ratio ,Qf-said;mli 1ip1iersi-of said terms and the group of components of the termsoisaid mathefimeeanstants .Qtsaid networks are proportioned mat ca e es so that. upon application of Isaid with respect 410' each other in the. ratio of .said file ri siena to a n works-thereane develexponents or said terms so that, upon applicaoped therein electrical effects varying as' predeiiiion of said electrical signal thereto, there is ,termined time functions'whioh at an instant of go developed in induet'orin each thereof a difs time determined by an independent-variable '-fQBli--V6ltag8=iiaryi11g2asa predetermined time known parameter of said mathematical series if oliionwhichataninstantof time determined reach in individual ones of said networks values is. n-independent-variableknown p m r of representative of different ones or the terms of aid m thematical series. reaches a. value represaid mathematical series; ,a'source of potential sentinel-one of the termslofsaid mathematical having ,a value representative of said in'dependseries: a source of potential having a value repent-variable known parameter; means jointlyreresenta't-iwe of said'independent-variable known sponsive to said potential of said source and at parameter;- means for utilizing said potential of least one of said electrical effects at said insaid source and at least one of said voltages at stant of said time for developing .a control effect; 35 said instant of said time to develop a control and a signalr-sampling circuit responsive to, said effect;- and a signal-jsamplmg circuit coupled to control efiect and coupled to at lea-st oneof said said utilizing means and to at least one of said networks for determining the value of said elecnetworks and responsive. to said control effect for :trical reflects developed therein at. said instantof determining the value of said voltages developed said time, thereby to derive :a resultant electrical .59 therein at said instant of said time to derive a .efiectrepresentative of the value of an unknown resultant voltage. representative of the value of parameter-of said mathematical series. 7 an; unknown parameter of said mathematical 9. An electrical computer forsolving equations series. involving mathematical power series including 1 1i An'electneal computer forsolving equaknqw n andunkngwnrpara meters and terms haw; 4; M0135 involving mathematical series having at ing coefficients and exponents comprising-2 means east sine .or' cosine terms and including known for supplying an electrical signal; a pluralityaof and unknown parameters; a-plurality of resonant resistorr-condenser networks at least some vof network eachhavin i p u p Circuits. which are electrically coupled in series relation arran ed in a. plurality of groups at least the with said supplying means; and the magnitudes nput circuits oi the networks of at least one of o the resistors and oondensersoi said network saidw unsioe ne f flQ lifial'ly coupled in series items so proportione that the impedanees? of relation-sad 'at leastgtheginput circuits of the said networks are proportioned wan-respects; networks; or another of said groups being elec- QdQIi other in the ratio of the coeflicients of the. tricjally independent of each other; means for terms of said mathematical; series-and thev time energizing said'networks coupled in series and constants of said networks are. proportioned, with for energizing: mdiv idnal ones of said networks respect to each other in the ratio of the expo: insaidanother of said groups; said networks wo ks. here are developed herein voltages ve s to nents of the terms of said series so that; upon each having circuit elements so proportioned applic tion o said. e ectrical signal to said-fillets th t. the-' -moedanees of. said; networks are proioned, respeetto each other in the ratio ing asex onentiallydecayin -time functions of anemones-oich stnut oi components oi: which. at an instant of time determined'hy an the torment-said mathematical series and the independent-variable known parameter of-"said resonant frequencies oi sagid networks are proinathematical series reach in individua'lc'nes-oi mriioned with resp ct to-eaeh-other in the ratio said networks values representative of different as f ndi idua 0m f an her roup f compoones of the terms of said mathematieal'series: a Items Of the terms of said-mathematical series source of potential having a value representa-r so thatnponbeing-energized there is developed tilve of. an in epen ent v riame kae n'na am in' -each ei aidn works n. oscillating si nal 'mean'sdointly responsive to said'petential whiehg-atan instant of time'- determined by an 1 i e endent-Wanamakn'own parameter of said @31 .m flfi alserienread s. a value represents 3 nd-6$lih6f fi8rfn$ Of I Iid. 'inathe-m ati-eal sesv i I1 i doul-flf-li tntialihaving a value represaid networks and responsive to saidcontrol ef-i sentative of said independent variable known said source and at least one of said oscillating signals at said instant of said time to develop a control effect; and a signal-sampling circuit coupied to said utilizing means and to the output circuit of at least one of said networks and re sponsive to said control effect for determining the value of at least one of said oscillating signals at said instant of said time to derive an electrical effect representative of the value of an unknown parameter of said mathematical series.

12. An electrical computer for solving equations involving mathematical series-having at least sine or cosine terms and including known and unknown parameters; a plurality of resonant networks each having input and output circuits, arranged in a plurality of groups at least the input circuits of the networks of at least one of said groups being electrically coupled in series relation and at least the input circuits of the networks of another of said groups being electrically independent of each other; means for energizing said networks coupled in series at an instant of time and for energizing individual ones of said networks in said another of said groups at the same instant of time; said networks each having circuit elements so proportioned that the impedances of said networks are proportioned with respect to each other in the ratio of individual ones of one group of components of the terms of said mathematical series and the resonant frequencies of said networks are proportioned with respect to each other in the ratio of individual ones of another group of components of the terms of said mathematical series so that upon being energized there is developed in each of said networks an oscillating signal which, at an instant of time determined by an independent-variable known parameter of said mathematical series, reaches a value representing one of the terms of said mathematical series; a source of potential having a value rep resentative of said independent-variable known parameter; means for utilizing said potential of said source and at least one of said oscillating signals at said instant of said time to develop a control effect; and a signal-sampling circuit coupled to said utilizing means and to the output circuit of at least one of said networks and re spcnsive to said control effect for determining the value of at least one of said oscillating Signals at said nstant of said time to derive an electrical efifect representative of the value'of an unknown parameter'of said mathematical series.

13. An electrical computer for solving equations involving mathematical'series having sine and cosine terms and including known and un-. known parameters; a plurality of resonant networks each having input and output circuits, arranged in a plurality of groups at least the input circuits of the networks of at least one ci's'aid groups being electrically coupledinserieskrelae tion and at least the input circuitsof the networks of another of said groups being electricalh independent of each'other; means for energiz ing said networks coupled in series and for energizing individual ones of said networks in said another of said groups; said networks each having circuit elements so proportioned that the impedances of said networksare=proportioned with respect to eachother in the ratio ofindividual ones of one group of components of the terms of.

said mathematical series and they resonant-fro quencies of said. networks areproporti'oned with. respect to each other in the ratio of individual ones of another group of components of the terms of said mathematical series so that upon being value representing one of the terms of said mathematical series; a source of potential having a value representative of said independentvariable known parameter; means for utilizing 'said'potential of said source and at least one of said oscillating signals at said instant of said time to develop a control eiiect; and a signal sampling circuit coupled to said utilizing means and to the output circuits of at least one of said networks and responsive to said control effect for determining the value of at least one of said oscillating signals at said instant of said time to derive an electrical effect representative of the value of an unknown parameter of said mathe matical series.

14. An electrical computer for solving equations involving mathematical series having terms including sine or cosine components and coeiiicient components; a plurality of resonant networks each having input and output circuits,

arranged in a plurality of groups at least the input circuits of the networks of at least one of said groups being electrically coupled in series relation and at least the input circuits of the networks of another of said groups being electrically independent of each other; means for energizing said networks coupled in series and for energizing individual ones of said networks in said another of said groups; said networks each having circuit elements so, proportioned that the impedances of said networks are proportioned with'respect to each other in the ratio of said coefilcient components and the resonant frequencies of said networks are proportioned with respect to each other in the ratio of the angles of said sine and cosine components of said terms so that upon being energized there is developed in each of said networks an oscillating signal having a frequency peculiar to the developing network and which, at an instant of time determined by an independent-variable known parameter of said mathematical series, reaches a value representing one of the terms of said mathematical series; a source of potential hav ing a value representative of said independentvariable known parameter; means for utilizing said potential of said source and at least one of said oscillating signals at said instant of said time to develop a control effect; and a signalsampling circuit coupled to said utilizing means and to the output circuit of at least one of said networks and responsive to said control effect for determining the value of at least one of said oscillating signals at said instant of said time to derive an electrical effect representative of the value of an unknown parameter of said mathematical series.

15. An electrical computer for solving equations having known and unknown parameters and involving mathematical series, including terms each having a sine or cosine function and a'multiplier, comprising: a plurality of resonant networks each having input and output circuits, arranged in a plurality of groups at least the input circuits of the networks of at least one of said groups being electrically coupled in series relation and at least'the input circuits of the networks of another of said groups being electrically independent of each other; means for energizing said networks coupled in series and for energizing individual ones of said networks in said another of said groups; said networks each including at least inductively coupled inductors and having circuit elements including said inductorsso proportioned and arranged that the coefficients of coupling of said coupled inductors are proportioned with respect to each other in the ratio of said multipliers of said terms and the resonant frequencies of said networks are proportioned with respect to each other in the ratio of the angles of said sine and cosine functions so that upon being energized there is developed in each of said networks an oscillating signal having a frequency and magnitude peculiar to the developing network and which, at an instant of time determined by an independent-variable known parameter of said mathematical series, reaches a value representing one of the terms of said mathematical series, said resonant characteristics of said networks being related to each other in the ratio of the angles of the sine or cosine of said terms and said coeflicients of coupling being related to each sponsive to said control effect for determining the value of at least one of said oscillating sig nals at said instant of said time to derive an electrical eifect representative of the value of an unknown parameter of said mathematical series.

CHARLES J. HIRSCH.

References Cited in the file of this patent UNITED STATES PATENTS Name Date Goldberg June 26, 1951 OTHER REFERENCES The Theory of Mathematical Machines; F. J. Murray; 2nd ed. 1948; Kings Crown Press; pages III-18, c

Number 

